Changes

So, the components of the above vector could be written <math>\vec{\boldsymbol{x}}=(x^1,x^2,x^3)</math>.

So, the components of the above vector could be written <math>\vec{\boldsymbol{x}}=(x^1,x^2,x^3)</math>.

But doesn't writing the vector components as superscripts lead to ambiguity with raising something to a power?  Not really.  Proper tensor/vector equations '''almost never raise a vector component to a power.'''  When one must do so, the convention is to put the power in parentheses.  So <math>x^2</math> is the second component of a vector, <math>x^{(2)}</math> is the number x squared, and <math>x^{2^{(2)}}</math> is the second component squared.

But doesn't writing the vector components as superscripts lead to ambiguity with raising something to a power?  Not really.  Proper tensor/vector equations '''almost never raise a vector component to a power.'''  When one must do so, the convention is to put the power in parentheses.  So <math>x^2</math> is the second component of a vector, <math>x^{(2)}</math> is the number x squared, and <math>x^{2^{(2)}}</math> or <math>(x^2)^2</math> is the second component squared.

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==Einstein Summation Convention==

==Einstein Summation Convention==

Index notation greatly simplifies the manipulation of [[tensor]]s. A commonly used convention when dealing with tensors is the so called "Einstein summation convention." Any index that is repeated exactly twice is assumed to be summed over and the sum symbols are not written.<ref>[http://mathworld.wolfram.com/EinsteinSummation.html Einstein Summation Convention at mathworld.wolfram.com]</ref> In the example above, the sum:

Index notation greatly simplifies the manipulation of [[tensor]]s. A commonly used convention when dealing with tensors is the so-called "Einstein summation convention." Any index that is repeated exactly twice is assumed to be summed over and the sum symbols are not written.<ref>[http://mathworld.wolfram.com/EinsteinSummation.html Einstein Summation Convention at mathworld.wolfram.com]</ref> In the example above, the sum:

:<math>

:<math>